Lagrangian distributions on asymptotically Euclidean manifolds

We develop the notion of Lagrangian distribution on a scattering manifold X. The latter is a manifold with boundary, with the boundary being viewed as points “at infinity.” In analogy with the classical case, a Lagrangian distribution is associated with a submanifold of the compactified cotangent bundle of X. The submanifold is Lagrangian with respect to a symplectic structure induced by the scattering geometry of X. Our analysis relies on the parameterization properties of the submanfold by means of local phase functions, and the study of the maps which preserve the scattering structure. We study the principal symbol map associating Lagrangian distributions with sections of a line bundle over the submanifold. In particular, we establish the principal symbol short exact sequence.